Relaxation times in an anharmonic crystal with diluted. impurities. Lapo Casetti. Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy

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1 Relaxation times in an anharmonic crystal with diluted impurities Lapo Casetti Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy Roberto Livi y Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, Bologna, Italy Alessandro Macchi z Department of Physics, University of California, Irvine, California Marco Pettini x Osservatorio Astrosico di Arcetri, Largo E. Fermi 5, Firenze, Italy (July 16, 1995) Abstract Molecular dynamics calculations performed on a model of a Xe solid with diluted impurities made of I 2 molecules indicate the existence of a Crossover Energy, below which the time to reach thermodynamic equilibrium increases rapidly. This eect is associated with the existence of long-living out-ofequilibrium states typical of many degrees of freedom Hamiltonian systems at low energies. The possibility of an experimental verication of this eect by laser spectroscopy techniques is also discussed. PACS: b; i; Fi Europhysics Letters 32, 549 (1995) 1 Typeset using REVTEX

2 A very rapid increase of the time scales to reach thermodynamic equilibrium is a typical feature of many degrees of freedom Hamiltonian systems when the energy is lowered beyond some threshold value. This is a general result that has been mainly derived by numerical investigations on the dynamics of chains of nonlinearly coupled oscillators, starting with far from equilibrium initial conditions [1]. Various dynamical indicators agree on the determination of the energy value at which the crossover between slow and fast relaxation to thermodynamic equilibrium occurs [2]. This is a model dependent quantity, which, from now on, we shall refer to as Crossover Energy (CE). From a physical point of view the relevant problem is the dependence of CE on the number of particles N. In particular, if CE would vanish in the thermodynamic limit (as one might expect on the basis of naive statistical arguments) the eect of the dynamics on the relaxation time scales would be of no practical interest for a real macroscopic system, e.g. an anharmonic crystal. Numerical simulations of anharmonic chains with cubic or quartic leading nonlinearity show that CE does not change signicantly when N is varied over two orders of magnitude (typically from 32 to 2048 particles) [3]. Let us stress that for both classes of potentials the estimated value of CE corresponds to small nonlinearities [4]. This implies that CE is determined by the leading anharmonic term in the Taylor series expansion of the potential, almost independently of its analytic form (provided the parameters of the dierent potentials in each class are rescaled to the same units). Lyapunov analysis for such classes of models indicates that the presence of CE is related to a transition from weakly to strongly chaotic dynamical regimes [5]. The persistence of CE in the thermodynamic limit is conrmed by the analytic estimate of the maximum Lyapunov exponent in the 1-d Fermi-Pasta-Ulam -model [6,7]. Since most of these results have been obtained for 1-d models, a further relevant question concerns the dependence of CE on the space dimension. Combining the numerical results obtained for oscillators coupled by the Lennard-Jones (LJ) 6? 12 potential in 1-d [8], 2-d (square lattice) [9] and 3-d (fcc lattice) [10], it is reasonable to conclude that CE is also independent of the space dimension. This scenario suggests that the existence of dierent time scales to reach thermodynamic 2

3 equilibrium as a consequence of a classical dynamical eect should be susceptible of experimental verication on a macroscopic system. This Letter aims at studying the feasibility of such an experiment [11]. In view of the above mentioned results, a rare gas solid, which can be reliably modeled as a monoatomic crystal with LJ 6? 12 interaction potential between neighbouring atoms, is expected to exhibit all the interesting mechanisms of relaxation to thermodynamic equilibrium that we want to single out. It is worth to observe that numerical studies are usually performed considering far from equilibrium initial conditions in the form of wave-packet excitations [1{3,5]. Unfortunately, it is practically impossible to produce such a kind of ballistic excitations in a pure atomic crystal. In order to circumvent this diculty, we consider the realistic case of a fcc crystal of Xenon atoms with diluted impurities of Iodine molecules. More precisely, we consider the case of impurity concentrations small enough ( 10?3 ) to prevent the formation of molecular clusters in a real sample [12]. Such a kind of crystal can be grown by usual cryogenic techniques from a gaseous mixture of Xe and I 2. The reason for introducing diluted I 2 impurities is that one can easily produce far from equilibrium initial conditions under the form of localized thermal excitations. Actually, energy can be stored instantaneously (at least with respect to the typical time scales that we are interested to investigate) in a vibrational level of the I 2 electronic ground state by a nonresonant stimulated Raman excitation [13]. At this instant of time the energy stored in the I 2 vibrational spectrum starts to be transferred to the phonons of the atomic crystal, eventually yielding thermalization of the whole sample. Notice that the diluted impurities do not modify signicantly the lattice structure of the Xe crystal, since the interaction potential between Xe and I atoms is very close to the Xe-Xe one. Actually, this interaction potential can be reasonably approximated by the average between the LJ 6? 12 potentials acting between neighbouring Xe atoms and between neighbouring I atoms of dierent molecules in an Iodine molecular crystal [14]. More precisely, these interaction potentials have the form 3

4 a = 4" a " a r 12? a r 6 # ; a = 1; 2 (1) where r is the distance between interacting atoms, " a is the depth of the potential well and a is the eective atomic radius. The index a = 1; 2 has been introduced as a shorthand notation for the Xe-Xe and Xe-I cases, respectively. The values of the physical parameters are " 1 = 3:1610?14 erg, " 2 = 4:3610?14 erg, 1 = 3:92 A, and 2 = 1. The intramolecular potential acting between the atoms of an I 2 molecule is of the Morse type M = fexp[?2(r? r 0 )]? 2 exp[?(r? r 0 )]g ; (2) where, due to the high dilution of I 2, one can use the free molecule values = 2:43 10?12 erg, = 1:52 A?1 and r 0 = 2:67 A. Notice that in these physical units the depth of the Morse potential well is roughly two orders of magnitude greater than the one of the LJ 6?12 cases. Our rst goal is to locate the CE of this system on a physical scale. In particular, we want to verify that CE appears at suciently high temperatures so that quantum eects can be neglected. For this purpose we have performed molecular dynamics calculations at constant pressure for a Hamiltonian system of 7 elementary cells per edge of the fcc lattice (corresponding to 1372 atoms) with periodic boundary conditions. One I 2 molecule is positioned at a site of the fcc crystal, replacing one Xe atom. Such a sample can be thought as an elementary cell of an innite crystal with diluted I 2 impurities located on a sublattice. This geometry cannot be obtained in a real sample by any crystal growth technique. Nevertheless, the dynamical properties of the thermal excitations induced on the lattice by exciting a vibrational state of the I 2 molecule seem quite independent of the geometry of the impurities, as we shall discuss later. The initial conditions are chosen in the following way. We rst use a MonteCarlo algorithm to prepare an initial conguration corresponding to a thermal equlibrium state of the whole sample at a given temperature T 0. The far from equilibrium initial condition is then obtained by adding a vibrational excitation to the I 2 molecule. In practice we have changed 4

5 the momenta of both atoms in the I 2 molecule by a proper amount, so that when the system eventually reaches thermal equilibrium the temperature of the system is T s = T 0 + T e [15]. The dynamical evolution of the sample is then studied by a bilateral symplectic algorithm [16], which guarantees a very good conservation of the energy (up to 0.01% over the explored range of energies using an integration step t = 8:25 10?15 s). An ecient probe to locate CE and to estimate the relaxation time R to thermodynamic equilibrium is the time dependent function (t) = 1? S(t) S max ; (3) where S(t) =? NX i=1 p i (t) log p i (t) ; (4a) p i (t) = T i(t) T tot (t) ; (4b) and T i (t) is the kinetic energy of the i-th particle, while T tot (t) = P N i=1 T i(t) [17]. In units of Boltzmann's constant T i (t) represents the \temperature" of the i-th particle; the maximum value of S(t), S max = log N, is obtained if all particles have the same temperature. We have used S max as a normalization factor for the expression of (t). Fluctuations at thermodynamic equilibrium prevent (t) from vanishing exactly. In fact, Monte-Carlo estimates of its equilibrium value give eq = 0:041. Notice that this value is independent of T s, as can be immediately proved using the denition of statistical ensemble average. Starting from the chosen initial conditions the typical evolution of (t) is rst characterized by a fast decrease, followed by a complex uctuating regime, whose features and duration depend on T s. After this transient, during which (t) maintains larger than eq, the relaxation towards eq sets in almost monotonously, thus yielding an unambiguous determination of R. A quantitative conrmation of these results has been obtained by measuring also the time needed for the Xe matrix and for the I 2 molecule to reach the equilibrium temperature T s. The relaxations times obtained with the two methods agree within a numerical factor of few units. 5

6 In Fig. 1 we report R as a function of T 0. We can conclude that CE is located around T 0 = 30 K, a value which is high enough to guarantee the reliability of a classical description for the relaxation process under investigation. It is worth to observe that, above CE, R is weakly dependent on T 0, implying that for suciently high temperatures of the crystal the mechanism of relaxation to thermodynamic equilibrium is ruled by a strongly chaotic regime. On the other hand, the sharp increase of R below CE can be interpreted as a consequence of the weakly chaotic dynamics of the lattice [5]. More precisely, below and above CE the dynamics resulting from the chosen initial conditions exhibits very dierent mechanisms yielding the thermal equilibrium state. In particular, for T 0 smaller than CE the excitation of the I 2 molecule is transmitted to the atoms by localized thermal waves, that, at early times, propagate almost unchanged through the lattice with constant velocity. Exploiting the periodic boundary conditions we have analyzed the interaction of such waves among themselves and with the molecule. The scattering eects are very weak and it takes some time before the travelling waves are denitely destroyed by nonlinear interactions with the lattice oscillations. For T 0 larger than CE the thermal uctuations of the lattice have large enough amplitude to destroy so rapidly the localized thermal waves that these can hardly interact among themselves and with the molecule. A more rened and quantitative analysis about the nature of these excitations will be reported elsewhere [18]. Nonetheless, these qualitative remarks indicate that the location of diluted impurities in the Xe crystal has little inuence on the thermalization mechanisms in both dynamical regimes. In this sense our model is expected to reproduce quite condently the properties of a macroscopic sample. Unfortunately, (t) is not a quantity accessible to experimental measurements. Let us observe that in an experimental set-up a thermostat has to be used to prepare the crystal at the equilibrium temperature T 0. On the other hand the excitation of the I 2 molecules will rise the equilibrium temperature of the crystal to T s over a time R. Notice that, even below CE, R is suciently small with respect to the typical response time of a thermostat, so that during this time the interaction of the crystal with the thermostat can be neglected. On 6

7 the other hand, this implies that also direct thermometric measurements are unaccessible in this case. A quantity that can be measured in an experiment by laser spectroscopy is the autocorrelation function of the velocity v of an atom in the I 2 molecule, C(t) = hv( + t)v()i hv()i 2? 1 : (5) By sampling in time the Fourier transform of this quantity, i.e. the associated power spectrum C(!), one can reconstruct the energy ow from the I 2 molecule to the acoustic band of the crystal. In order to verify that this method conveys the same information as (t)- analysis does, we have computed also C(!) by our molecular dynamics simulations. In Fig. 2 we show snapshots of C(!) at dierent times for T 0 = 10 K and T 0 = 50 K. We nd that the two methods are in agreement [19], thus showing that an experimental determination of R should be possible. The outcomes of the outlined experiment might then provide the rst experimental evidence of the existence of CE in a macroscopic sample. In conclusion, we have shown that long-living out-of-equilibrium states, typical of weakly chaotic classical dynamics, are expected to produce measurable eects in a real macroscopic system. We want to thank S. Califano, G. Di Tocco, P. Foggi and R. Righini for useful discussions and suggestions. E.G.D. Cohen, H.A. Posch and A. Tenenbaum are warmly acknowledged for advice and criticism. One of us (A.M.) wishes to thank A.A. Maradudin for useful discussions. We are indebted with CECAM in Lyon for its kind hospitality during the workshop \Chaotic and ordered energy ow in lattices" where part of this work was done. 7

8 REFERENCES Also at INFN, Sezione di Firenze, Largo E. Fermi 2, Firenze, Italy. Electronic address: y Also at INFN, Sezione di Bologna, via Irnerio 46, Bologna, Italy, and INFM, Unita di Firenze, Largo E. Fermi 2, Firenze, Italy. Electronic address: z On leave from Dipartimento di Fisica, Universita di Firenze, Largo E. Fermi 2, Firenze, Italy. Electronic address: x Also at INFN, Sezione di Firenze, and INFM, Unita di Firenze, Largo E. Fermi 2, Firenze, Italy. Electronic address: [1] P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger, Phys. Rev. A 2, 2013 (1970); M. Casartelli, G. Casati, E. Diana, L. Galgani, and A. Scotti, Theor. Math. Phys. 29, 205 (1976); R. Livi, M. Pettini, S. Ruo, M. Sparpaglione, and A. Vulpiani, Phys. Rev A 28, 3544 (1983). [2] R. Livi, M. Pettini, S. Ruo, M. Sparpaglione, and A. Vulpiani, Phys. Rev A 31, 1039 (1985); R. Livi, M. Pettini, S. Ruo, and A. Vulpiani, Phys. Rev A 31, 2740 (1985); M. Casartelli and M. Sello, Phys. Lett. A 112, 249 (1985). [3] H. Kantz, R. Livi and S. Ruo, J. Stat. Phys.76, 627 (1994). [4] In these models the CE is located at an energy scale where the contribution of the anharmonic terms in the Hamiltonian is only a few percents of the total energy. [5] M. Pettini and M. Landol, Phys. Rev. A 41, 768 (1990); M. Pettini and M. Cerruti- Sola, Phys. Rev. A 44, 975 (1991). [6] L. Casetti, R. Livi and M. Pettini, Phys. Rev. Lett. 74, 375 (1995). [7] L. Casetti and M. Pettini, Phys. Rev. E 48, 4320 (1993). [8] L. Casetti, G. Di Tocco, R. Livi, M. Pettini, and M. Spicci, in preparation. 8

9 [9] G. Benettin and A. Tenenbaum, Phys. Rev. A 28, 3020 (1983). [10] L. Casetti and A. Macchi, in preparation; L. Casetti, Tesi di Laurea, Universita di Firenze, (1993). [11] An experiment inspired by the numerical studies reported in this Letter is going to be performed at LENS in Firenze, Italy by S. Califano, G. Di Tocco, P. Foggi and R. Righini. [12] M. L. Klein (Ed.), Inert Gases: Potentials, Dynamics, and Energy Transfer in Doped Crystals (Springer-Verlag, Berlin, 1984); W.F. Howard and L. Andrews, J. Raman Spectr. 2, 447 (1974). [13] A. Lauberau and W. Kaiser, Rev. Mod. Phys. 50, 607 (1978). [14] K. Kobashi and R.D. Etters, J. Chem. Phys. 79, 3018 (1983). [15] This means that the vibrational energy initially stored in the molecule is Nk B T e. (In our simulations Nk B T e 0:25 ev). [16] L. Casetti, Physica Scripta 51, 29 (1995). [17] This quantity is analogous to the spectral entropy used in Refs. [2,3] for wave-packet excitations. [18] G. Di Tocco, in preparation. [19] The relaxation times computed by means of C(!) exceed those computed by means of (t) by a factor of two. This is not surprising because it is well known that dierent observables can have dierent relaxation times. 9

10 FIGURES FIG. 1. Relaxation times R as obtained from the decay of (t) at varying initial bulk temperature T 0 and xed excitation. Errorbars measure the uncertainty in the determination of R due to the uctuations of (t). All the simulations have been performed using a time step t = 8:2510?15 s and an excitation temperature T e ' 1:5 K. FIG. 2. Time behaviour of the power spectrum C(!) of the velocity of the I 2 molecule - obtained by molecular dynamics simulations - at T 0 = 10; 50 (upper two plots). C(!) is measured in arbitrary units and! in natural Xe units Hz. The rst ten slices in each plot are obtained by averaging during temporal windows - centered around the t values reported in gure - and of 60 ps width. The last slice, superimposed on the!-axis of each plot, is computed averaging over equilibrium congurations at T s, generated by a MonteCarlo algorithm. The energy ow from the molecular vibrations towards the phonons of the Xe matrix is monitored by the time behaviour of the intensity I ph in the low-frequency (! < 50 in natural units) part of the spectrum, which is plotted against time in the lower plot for T 0 = 10 K (solid circles) and T 0 = 50 K (solid triangles). The values of t and T e are again the same of Fig